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Thermal Process Calculations Through Ball’s Original FormulaMethod: A Critical Presentation of the Method and Simplificationof its Use Through Regression Equations

Nikolaos G. Stoforos

Received: 10 August 2009 / Accepted: 8 February 2010 / Published online: 3 March 2010

� Springer Science+Business Media, LLC 2010

Abstract Ball’s formula method is a classical method

used in the thermal processing industry, the basis and the

precursor of a number of most recent methodologies. It has

received a lot of attention, being reviewed, criticized,

compared and evaluated by several investigators. The use

of Ball’s method relies on appropriate diagrams which

sometimes are difficult to use, find or reproduce. After

presenting the principles involved in the development of

Ball’s formula method, reviewing the literature related to

that particular method, and discussing its limitations, some

working, regression equations developed in order to facil-

itate the application and the use of the method are

presented.

Keywords Ball’s formula method � Thermal processing �Canned foods � Process calculations

Latin Letters

a1–a7 Regression coefficients appearing in Eq. 27,

dimensionless

b1–b7 Regression coefficients appearing in Eq. 28,

associated with temperature differences, g and z,

in �F

B Steam-off time (measured from corrected zero),

min

C Concentration of a heat-labile substance, number

of microorganisms/mL, spores per container,

g/mL, or any other appropriate unit

CUT Duration of retort come-up time, min

DT

(Noted also as D) decimal reduction time or death

rate constant; time at a constant temperature

required to reduce by 90% the initial spore load

(or, in general, time required for 90% reduction of

a heat-labile substance), min

E1 Exponential integral defined by Eq. 15

Ei Exponential integral defined by Eq. 16

FzT (Or simply F) time at a constant temperature, T,

required to destroy a given percentage of

microorganisms whose thermal resistance is

characterized by z, or, the equivalent processing

time of a hypothetical thermal process at a

constant temperature that produces the same

effect (in terms of spore destruction) as the

actual thermal process, min

Fi Factor defined by Eq. 14, which when multiplied

FTref by gives the F value at the retort temperature,

dimensionless

f Time required for the difference between the

medium and the product temperature to change by

a factor of 10 min

g Difference between retort and product temperature

(at the critical point) at steam-off time, �F

g0 A small g value lower than or close to 0.1�F, after

which product temperature is considered constant, �F

I Difference between retort and initial product

temperature, �F

j A correction factor defined by Eqs. 5 and 8 for the

heating and cooling curve, respectively, based on

the intercept, with the temperature axis at time

zero, of the straight line that describes the late,

straight, portion of the experimental heating or

cooling curve plotted in a semi-logarithmic

temperature difference scale as shown in Figs. 1

and 2, respectively, dimensionless

N. G. Stoforos (&)

Department of Food Science and Technology, Agricultural

University of Athens, Iera Odos 75, 11855 Athens, Greece

e-mail: [email protected]

123

Food Eng Rev (2010) 2:1–16

DOI 10.1007/s12393-010-9014-4

m Difference between product temperature at steam-

off and retort temperature during cooling, defined

by Eq. 12, �F

T (Product) temperature, �F

TA Extrapolated pseudo-initial product temperature at

the beginning of heating defined as the intercept,

with the temperature axis at time zero, of the

straight line that describes the late, straight,

portion of the experimental heating curve plotted

as shown in Fig. 1, �F

TB Extrapolated pseudo-initial product temperature at

the beginning of cooling defined as the intercept,

with the temperature axis at zero cooling time, of

the straight line that describes the late, straight,

portion of the experimental cooling curve plotted

as shown in Fig. 2, �F

Th Product temperature at the beginning of the

cooling cycle, �F

t Time, min

U The F value at TRT, defined by Eq. 13, min

u Dummy variable

z Temperature difference required to achieve a

decimal change of the DT value, �F

zc A correction temperature difference factor

appearing in Eq. 27, �F

Latin Letters

q The fraction of the total lethal value of a process (that is,

the F value of the entire process) which is achieved during

the heating cycle only of the thermal process, assuming

that the slope of the heating curve is constant and equal to

the slope of the cooling curve, dimensionless

Subscripts

1, 2 Refers to a particular condition

a Initial condition

b Final condition

bh Condition at the time where the break, the

change in the slope of the heating curve occurs

CW (Water) cooling medium

c Cooling phase

end End of cooling cycle

g Condition at steam-off time

h Heating phase

IT Initial condition (for product temperature only)

process Referring to process values

RT (Retort) heating medium

ref Reference value

required Referring to required values

Introduction

Thermal process calculations refer to the design and/or the

evaluation of a thermal process in terms of its ability to

extend the shelf life of a product by destroying undesirable

agents present in the food while preserving the quality

characteristics of the product. The General Method [9] was

the first method developed for thermal process calculations.

It was the starting point for a scientific approach in the

production of canned products, and it set the fundamental

principles for subsequent developments.

The General Method is an exact method as far as the

evaluation of a thermal process is concerned, but it lacks

the predictive power needed for design purposes. The

method developed by C. Olin Ball and at first presented in

1923 not only made the tedious calculations required by the

General Method easier, but it successfully engineered the

thermal process design task. It was the first of the so called

Formula Methods, a classical method still used in the

thermal processing industry, recognized as ‘‘a major

milestone in the history of food technology’’ by Merson

et al. [31] in their evaluation of the method. It served as the

model for a number of methods, some of them directly

influenced being those by Ball and Olson, Herndon et al.,

Griffin et al., Hayakawa, Larkin, and Larkin and Berry

[6, 23, 18, 19, 20, 28 and 30]. Limited comparisons of

Heating Curve (T RT =250°F)

210

220

230

240241242243244245

246

247

248

249

200190180170160150

50

-50

-150

-250-350-450-550-650-750

TIT I=TRT-TIT

TA jhI=TRT-TA

0 20 40 60 80 100 120

Time (min)

Prod

uct T

empe

ratu

re, T

(°F

)

1

10

100

1000

Ret

ort -

Pro

duct

Tem

pera

ture

, TR

T -

T (

°F)

Exponential Fitting (to the "linear"portion of the experimental data)

Experimental Data

f h =48 min

2170

340 ==−−

=ITRT

ARTh TT

TTj

Fig. 1 Typical straight-line heating curve used for fh and jh parameter

estimation

2 Food Eng Rev (2010) 2:1–16

123

various formula methods have been reported in the litera-

ture [17, 42, 47].

In this presentation, we will be concentrated to matters

directly associated with Ball’s original formula method. The

term ‘‘original’’ formula method is used here in reference to

the method presented initially by Ball in 1923 and thereafter in

subsequent publications [4, 6], and to distinguish from a

‘‘new’’ method included in the 1957 publication [6]. Reviews

on mathematical procedures developed for thermal process

calculations are available [10, 21, 50]. A variety of comput-

erized thermal process calculation methodologies based on

numerical methods [27, 33, 55], and the General Method [41]

have been also appeared in the literature with some of them

being for commercial use [24]. They are flexible enough to

adjust to different process conditions as well as to handle

process deviations. A review of such methods, stretching their

advantages and limitations, will be always useful.

The objective of this work was to give a critical pre-

sentation of Ball’s original formula method for thermal

process calculations and to discus the assumptions involved

in terms of product safety and product quality. In order to

facilitate the use of the method, a set of regression equa-

tions was developed and is included in this article.

Theoretical Considerations

The concept of the F value of a thermal process is the

foundation of thermal process calculations. The F value of

a process is defined as the equivalent time of a hypothetical

thermal process at constant temperature, Tref, that produces

the same result, as far as the destruction of microorganisms

or other spoilage agents is concerned, with the actual

thermal process, during which product temperature can be,

and usually is, nonconstant. In the remaining of the text,

when talking about thermal destruction we will be referring

to microbial destruction, although the analysis will apply to

any heat-labile substance. From the aforementioned defi-

nition of the F value, and accepting first order kinetics for

microbial thermal destruction [12], we can further define

the F value as the time, at a constant temperature, Tref,

required to destroy a given percentage of microorganisms.

Following the classical thermobacteriological approach for

first order destruction kinetics [6, 32, 51], the F value can

be mathematically expressed as:

FzTref¼ DTref

ðlogðCaÞ � logðCbÞÞ ¼Ztb

ta

10TðtÞ�Tref

z dt ð1Þ

Superscript z on the F value indicates that the F value of

a process is associated to a particular type of

microorganism or heat-labile substance characterized by

a given z value. More details and some discussion

concerning the above equation can be found elsewhere

[49]. In brief, Eq. 1 suggests evaluation of the F value at a

particular point in the product either through microbial

concentration data at that particular point at the beginning

and at the end of the process, or through the time

temperature data observed at the point of interest

throughout the entire process. Although the two modes of

calculation produce equivalent results, provided the

parameters involved are known with adequate accuracy,

it is customary to use microbial reduction data to establish

a required F value (Frequired) and temperature evolution

data to calculate the F value of a given process (Fprocess).

There are two type of problems associated with thermal

process calculations. The first one refers to the calculation

of the Fprocess value of a given process; that is, a process for

which we know the heating time and the time–temperature

history of the product throughout the total processing time.

The second problem refers to the calculation of the

required heating time, at given processing conditions, in

order to achieve a target Frequired value; an F value which is

Cooling Curve (T CW =70°F)

110

100

90

8079787776

75

74

73

72

71

120

130140150160170

270

370

470

570

670770870970

1070

Th Th-TCW

TB TB-TCW

0 20 40 60 80 100 120

Time (min)

Prod

uct T

empe

ratu

re, T

(°F

)

1

10

100

1000

Prod

uct -

Coo

ling

Wat

er T

empe

ratu

re, T

- T

CW

(°F

)

Exponential Fitting (to the "linear"portion of the experimental data)

Experimental Data

f c =48 min

41.17.172

5.243 ==−−

=CWh

CWBc TT

TTj

Fig. 2 Typical cooling curve used for fc and jc parameter estimation

Food Eng Rev (2010) 2:1–16 3

123

based on predescribed microbial reduction requirements. If

we split the integral of Eq. 1 into two parts, the first one

referring to the heating cycle and the second one to the

cooling cycle (that inevitably follows the heating cycle) of

the thermal process, that is,

Ztb

ta

10TðtÞ�Tref

z dt ¼Zth¼B

th¼0

10TðthÞ�Tref

z dthþZtc¼tend

tc¼0

10TðtcÞ�Tref

z dtc ð2Þ

then, both types of problems can be attacked through the

solution of the same equation:

Rth¼B

th¼0

10TðthÞ�Tref

z dthþRtc¼tend

tc¼0

10TðtcÞ�Tref

z dtc

Fz

Tref

¼ 1 ð3Þ

If the heating time, B, of a given process is known, the

solution for Fz

Trefthrough Eq. 3 gives the Fprocess value. This

is the process evaluation step and, given the time–

temperature data of the product during the entire process,

it is rather straightforward. Alternatively, if Fz

Trefrepresents

the Frequired value, one can solve Eq. 3 for the heating time,

B, the upper limit of the first integral in the above equation.

This is the design step of a thermal process, a much more

difficult mathematical task, which also requires knowledge

of product time–temperature evolution data. As a matter of

fact, this step is the problematic step in the General Method

for thermal process calculations presented by Bigelow and

his colleagues in 1920 and which probably led Ball into the

development of his Formula Method. We must notice here

that an a priori knowledge of the F value that will be

accumulated after the end of the heating (i.e., during the

cooling cycle of the process) as it is represented here by the

second integral of Eq. 3 is a prerequisite for the calculation

of the heating time and thus for the process design step.

Ball’s Formula Method

As far as food safety, in terms of microbial destruction, is

concerned, Ball was a supporter of the single point F value

calculation theory. He reasoned that ‘‘bacterial spores are

most likely to survive in the unit volume which receives the

smallest amount of lethal heat’’ [5]. Thus, for conduction

heated foods for example, the geometric center of the

container or a region close to it [14, 54] represents the

critical point of the product; the point that is least affected

by the process. If the product at this point is ‘‘safe’’, then

the whole product is safe. Thermal process calculations, as

for example temperature data used with Eq. 3 for F value

calculations should be, therefore, based on the critical point

of the product.

The approach that Ball [3] used in developing his ori-

ginal method set the basis of a group of calculation

methods named Formula Methods [50]. He first developed

a set of equations to describe the temperature of the

product, at the critical point, as a function of processing

time and then, by substituting these equations into Eq. 3

and solving the resulted equation, he obtained a relation-

ship between heating time and F value. Actually, as it will

be presented later on, among a choice of independent and

depended variables, Ball expressed the results of integra-

tion of Eq. 3 through the dimensionless fh/U parameter and

the difference between retort and product temperature, at

the critical point, at the end of heating (g).

Based on observations of the temperature evolution at

the critical point of canned foods, Ball concluded that any

heating or cooling curve, when appropriately plotted, after

an initial lag was asymptote to one (straight-line curve) or

more (broken-line curve) straight lines. Such heating and

cooling curves, as traditionally plotted in thermal process

literature (for example, in ‘‘inverse’’ semi-logarithmic

retort minus product temperature difference scale, for the

heating curve) are shown in Figs. 1 and 2, respectively

(temperatures will be given in degrees Fahrenheit and time

in minutes, following the terminology in Ball’s publica-

tions and the units associated with the use of Ball’s

method). After the initial lag, each curve could be, there-

fore, approximated by its asymptote. So, for a straight-line

heating curve, he suggested the following equation:

TðthÞ ¼ TRT � jhðTRT � TITÞ10�th=fh ð4Þ

Equation 4 assumes constant retort temperature

throughout the heating cycle, and as it is evidence from

this equation, the entire heating curve was described

through two empirical parameters, the fh and jh values. The

fh value is related to the slope of the heating curve and it is

defined as the time required for the difference between

retort and product temperature to change by a factor of 10,

that is, in relation to Fig. 1, as the time needed for the

straight-line heating curve to traverse through a logarithmic

cycle. The jh value is a dimensionless lag factor [48]

defined as:

jh ¼TRT � TA

TRT � TITð5Þ

for TA being an extrapolated pseudo-initial product tem-

perature at the beginning of heating (Fig. 1). Equation 4

written as temperature difference ratios, that is, (TRT–T(th))

over (TRT–TIT) suggests the independency of the fh and jhvalues on the temperature units used.

Determination of the fh and jh parameters might intro-

duce some error in subsequent calculations, not necessarily

for their graphical determination as it was initially done

(computer software allows nowadays accurate data fitting

4 Food Eng Rev (2010) 2:1–16

123

and parameter estimation) but because their values are

affected by the length of heating time during which heat

penetration data available [34]. Nevertheless, these two

empirical fh and jh parameters can be related, through

theoretical expressions, to product properties and process

parameters, so that they can be transferred from a given

experiment to a number of different product and process

conditions [6, 34].

Ball used Eq. 4 for the entire heating curve, reasoning

that the thermal destruction taking place at the beginning of

a thermal process, where Eq. 4 might not successfully

describe product temperature data, is negligible since,

under common commercial practices, the temperature of

the product has not yet reached the lethal temperature

range. Contrary to this, the cooling curve was described by

two equations: one for the initial curved portion of the data

and another for the straight-line part of the curve. At the

beginning of cooling, product temperature is high and this

must be accurately taken into account in order to arrive to

correct process calculations. So, Ball used a hyperbola to

describe the initial part of the cooling curve for

0 B tc B fclog(jc/0.657), as given here by Eq. 6:

TðtcÞ¼Tgþ0:3ðTg

�TCWÞ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

0:5275logðjc=0:657Þ

� �2tc

fc

� �2s2

435

ð6Þ

where Tg is the temperature of the product at the critical

point at the end of heating, and TCW is the retort temper-

ature during the cooling cycle.

Based on experimental observations, Ball [3] made a

series of assumptions about the parameters defining the

hyperbola. He fixed the shape of the hyperbola through

various constants and parameters, which were based on the

difference between product temperature at the end of heating

(Tg) and the retort temperature during cooling (TCW).

For the straight-line portion of the cooling curve, that is, for

tc C fclog(jc/0.657), Ball used another exponential equation:

TðtcÞ ¼ TCW þ jcðTg � TCWÞ10�tc=fc ð7Þ

As for the heating cycle, the fc and jc parameters are

determined from the cooling curve (Fig. 2). In particular, jcis defined as:

jc ¼TB � TCW

Th � TCWð8Þ

In a next step, based on experimental observations and

conservative assumptions, Ball fixed the jc value at 1.41

and he further assumed fc = fh, being thus liberated from

the need of collecting cooling data. Under the new

assumptions, Eq. 6 is reduced to:

TðtcÞ ¼ Tg þ 0:3ðTg � TCWÞ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ tc

0:175fh

� �2s2

435

ð9Þ

Three main restrictions being so far imposed in Ball’s

formula method due to the assumptions made about

product temperature during the cooling cycle (Eq. 7 and

9) are as follows: First, the equation used to describe the

initial part of the cooling curve (Eq. 9) does not allow for

the product temperature to increase at the beginning of

cooling. This is not valid for the geometric center (or the

critical point) of conduction heated canned foods. Second,

a jc value of 1.41 can be rarely (and accidentally) true and

might be a severe limitation. A jc value of 1 is expected for

perfect mixing forced convection heated foods, while a

value of 2.04 is the theoretical value for conduction heated

canned foods at uniform initial temperature at the

beginning of cooling and processing conditions that result

in infinite heat transfer coefficient between product and

cooling medium. Third, the assumption about fc = fh is an

assumption that facilitates calculations. The case where the

slope of the heating curve is different from the slope of the

cooling curve (that is, the case when fc = fh) has been

discussed and accounted for by Ball [3]. The

aforementioned limitations will be further addressed in

this article.

Having described with the appropriate equations product

temperature evolution at the critical point for the entire

thermal process, Ball proceeded with the substitution of

these equations (Eqs. 4, 7 and 9) into Eq. 3 and, after

considerable mathematical operations, he ended up with

the following equation:

fh

U

1

lnð10Þ elnð10Þg=z

� elnð10Þg=z E1

lnð10Þ gz

� �� E1

lnð10Þ 80

z

� �� ��

þ 0:76381e�0:789m=z þ 0:5833z

me0:692m=zE

h i

þ e� lnð10Þm=z

�Ei

lnð10Þz

0:657m

� �

� Eilnð10Þ

zðmþ g� 80Þ

� ���¼ 1 ð10Þ

Each one of three terms in brackets in Eq. 10

corresponds to the heating cycle, the initial lag of the

cooling cycle, and the final, straight line, part of the cooling

cycle, respectively. The parameter g in Eq. 10 is the

difference between retort and product temperature, at the

critical point, at the end of heating,

Food Eng Rev (2010) 2:1–16 5

123

g ¼ TRT � Tg ð11Þ

while the parameter m is the difference between product

temperature at the end of heating and retort temperature

during cooling, that is,

m ¼ Tg � TCW ð12Þ

The parameter U is the F value at retort temperature,

TRT, that is,

U ¼ FzTref

Fi ð13Þ

for

Fi ¼ 10Tref �TRT

z ð14Þ

The exponential integrals, E1(x) and Ei(x) are defined as

[16]:

E1ðxÞ ¼Z1

x

e�u

udu ð15Þ

and

EiðxÞ ¼Zx

�1

eu

udu ð16Þ

Values of the above exponential integrals can be

obtained through various approximations of Eqs. 15 and

16 or from tables [16]. The quantity E appearing in Eq. 10

is a numerical evaluated integral defined as:

E ¼Zlnð10Þ0:643m=z

lnð10Þ0:3m=z

e�u

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 � ðlnð10Þ0:3m=zÞ2

qdu ð17Þ

There is an additional assumption hidden in Eq. 10. Due to

the limits of integration of Eq. 3, the initial product

temperature at the beginning of the heating cycle, as well as

the final product temperature at the end of the cooling cycle

should, somewhere, appear in Eq. 10. In order to have results

independent of these two temperatures, Ball assumed that

there is no lethal effect of the process for product temperatures

less than 80�F below the retort temperature during heating,

TRT. Thus, integration starts at TRT-80 (�F), giving the term

E1(ln(10)80/z) in Eq. 10, and ends up at when product

temperature reaches again TRT-80 (�F) during cooling, from

where the term Ei(ln(10)(m ? g–80)/z) in Eq. 10 comes

from. In consistence with all other temperature terms that

appear on Eq. 10 as differences between product and (heating

or cooling) retort temperature, note that the quantity m ? g–

80 is equal to TRT-80-TCW, that is the difference between

product temperature at the end of cooling (TRT-80) and retort

temperature during the cooling cycle (TCW).

At this point, we must mention that Eq. 10 is presented

here in its corrected form [46], which is slightly different

than the corresponding equations presented by Ball in his

first publication of the method [3] and by Ball and Olson in

their book on 1957 [6]. As a matter of fact, the equations

presented in these two publications, that is, in Ball [3] and in

Ball and Olson [6] are not only different than Eq. 10, but

they are also different between themselves [24, 46]. This

created some controversy [15, 43, 45] until finally being

resolved [29, 46]. Tabulated or graphically presented results

in all Ball’s publications dealing with the presentation of his

original formula method [3, 4, 6] are identical, but not in

accordance to the aforementioned two equations presented

in Ball [3] and Ball and Olson [6] publications [42]. Dis-

crepancies between Ball’s equation and the tabulated values

were attributed to typographical errors with the validity of

Ball’s tables being reestablished and the correct form of the

equation, as presented here by Eq. 10 being presented [46].

With the aforementioned remarks, and noting that Eq. 10

does not possess an analytical solution, after the graphical

evaluation of the integral E, Ball presented discrete values of

the solution of Eq. 10 in tabular or graphical forms [3, 4, 6].

His graphs or tables correlated fh/U values vs g or log(g)

having the z and the m ? g values as parameters. Ball pre-

sented results for three m ? g values, namely 130, 160 and

180�F, and 11 z values namely 6, 8, 10, 12, 14, 16, 18, 20, 22,

24 and 26�F. Such a graph, reproduced from the tabulated

values from [6], is presented here in Fig. 3. Note that Ball

used temperatures in degrees Fahrenheit which we kept in

this article too. Based on Eqs. 11 and 12, m ? g is defined as:

mþ g ¼ TRT � TCW ð18Þ

Minimum g values employed were equal to about 0.1�F

(log(g) = -1) as can be seen from Fig. 3 as well as Ball’s

graphs and tables [3, 4, 6]. For thermal processes with

g \ 0.1�F, Ball suggested to consider that the temperature

at the critical point remained constant and equal to TR -

0.1 (�F) from the moment the critical point has reached a

temperature of 0.1�F below retort temperature during

heating. Mathematically, for g \ 0.1�F, the correlation

between fh/U and log(g) is given by Eq. 19.

fhU

����g\0:1

¼100:1=z � fh

U

��g¼0:1

100:1=z � 1þ logðgÞð Þ � fhU

��g¼0:1

ð19Þ

where fhU

��g\0:1

is the fh/U value for the particular m ? g and

z value and for the g \ 0.1�F value of interest, and fhU

��g¼0:1

the corresponding fh/U value for g = 0.1�F, a value that

can be obtained directly from Ball’s graphs or tables for the

appropriate z value.

The above equation produces results absolutely equiv-

alent to the graphical data presented by NCA [32] in figures

where the log(g) range has being extended up to –8.

If another small enough g value, lower than or close to

6 Food Eng Rev (2010) 2:1–16

123

0.1�F, g0, instead of the g = 0.1�F value is used from Ball’s

tables, then Eq. 19 reduces to:

fhU

����g\g0¼

10g0=z � fhU

��g¼g0

10g0=z � 1þ logðgÞð Þ � fhU

��g¼g0

ð20Þ

As mentioned earlier, the goal in thermal process

calculations is to obtain a relationship between heating

time and F value. The F value is directly correlated with

the fh/U variable through Eq. 12. The parameter g, the

difference between retort and product temperature (at the

critical point) at steam-off time, used as the independent

variable, was related to the end of heating (steam-off) time,

B, through the following formula (originated from Eq. 4):

B ¼ fhðlogðjhIÞ � logðgÞÞ ð21Þ

for

I ¼ TRT � TIT ð22Þ

Thus, through the solution of Eq. 10, and Eqs. 13 and

21, a relationship between heating time, B, and F value is

established and thermal process calculations can be

performed. In brief, the use of Ball’s method for

determining the F value of a given process, involves the

following steps:

1. Determination of the fh and jh values from the

experimental heating curve

2. Calculation of log(g) from Eq. 21 and the experimen-

tal steam-off time, B

3. Finding the fh/U value, for the log(g) value calculated

in step 2, from Ball’s graphs (for example, Fig. 3) and

the appropriate m ? g and z values

4. Calculation of F value (from the fh/U value found in

step 3) through the following equation (originated

from Eq. 13 -the definition of U):

FzTref¼ fh

fhUFi

ð23Þ

The second type of problem, that is, estimation of the

required heating (steam-off) time B in order to achieve a

particular Frequired value, requires the same steps, with the

last three steps being followed in reverse order.

Specifically, it involves the following:

1. Determination of the fh and jh values from the

experimental heating curve

2. Calculation of the fh/U value from the fh value found in

step 1 and the given Frequired value, through Eq. 13

3. Finding the log(g) value, for the fh/U value calculated

in step 2, from Ball’s graphs (for example, Fig. 3) and

the appropriate m ? g and z values

4. Calculation of the required steam-off time, B from

Eq. 21 and the log(g) found in step 3

Changes in retort or initial product temperatures do not

impose any particular problem with straight-line heating

curves since the fh and jh values are not affected.

Discussion

During the presentation of Ball’s formula method, we make

notice of a number of assumptions and simplifications that

were made in the development of the method. In the next

paragraphs we will try to address them.

Broken-Line Heating

The assumption about straight-line heating, that is, con-

stant fh throughout the entire heating cycle was in fact

handled by Ball. By providing additional tables (and

working equations), he allowed calculations for broken-

line heating curves, that is, heating curves that could not be

adequately described by a single straight line but needed a

0.1

1

10

100

1000

-1 -0.5 0 0.5 1 1.5 2

log(g) (g in °F)

fh/U

z = 6°F

z = 10°F

z = 14°F

z = 18°F

z = 22°F

z = 26°F

Fig. 3 Curves giving the fh/U vs log(g) relationship, according to

Eq. 10, for m ? g = 180�F and various z values (based on Ball and

Olson’s [6] data)

Food Eng Rev (2010) 2:1–16 7

123

set of straight lines, each one with different slope, to be

described. For the case of a broken-line heating curve with

two straight lines of different slopes, fh1and fh2

(that is,

with one break), the following equation correlates the F

value with the fh/U parameter associated with Ball’s fh/U vs

log(g) graphs or tables:

fhU¼ fh2

FzTref

Fi þqbhðfh2

�fh1Þ

fhUjbh

ð24Þ

The fhU

��bh

parameter represents the fh/U value obtained

from Ball’s fh/U vs log(g) graphs or tables for g = gbh. The

parameter gbh is the temperature difference between

product (at the critical point) and retort temperature at

the time where the break, that is, the change in the slope of

the heating curve occurs. Similarly, qbh, is the q value for

g = gbh. During the calculations involved in solving

Eq. 10, the contribution of each phase of the process

(heating cycle, the initial lag of the cooling cycle, and the

final part of the cooling cycle) could be easily separated. So

Ball, based on these calculations, was able to estimate the

fraction, q, of the Fprocess value, which is achieved during

the heating cycle only of the thermal process, (assuming

fc = fh = constant). As earlier, for the fh/U vs g

relationship, Ball [3] initially presented graphs with

curves correlated q with g and later tables q vs g [6] for

a wider range of m ? g (130, 150, 160 and 180�F) and z

(6, 14, 18, and 26�F) values.

Thus, thermal process calculations for broken-line heat-

ing curves could be performed in an analogous way to

straight-line heating curves. The Fprocess value could be

directly calculated from Eq. 24, while the formula connected

heating time, B, with temperature difference at steam-off, g,

for broken-line heating curves, takes the form of:

B ¼ fh1logðjIÞ þ ðfh2

� fh1Þ logðgbhÞ � fh2

logðgÞ ð25Þ

Ball presented equations (similar to Eq. 24) considering

up to five different slopes (three for the heating curve and

two for the cooling curve) and in principle, one can extent

the methodology to handle any shape of heating and

cooling curves. This though is troublesome, subject to

errors, and probably the use of a General Method scheme is

more advisable.

In a last comment about broken-line heating curves, we

must mention that, contrary to straight-line heating curves

where changes in retort or initial product temperatures can

be easily handled, this is not the case for broken-line

heating. Since there is no method that provides a way of

estimating the time or the temperature where the brake

occurs for different processing conditions, it is considered

unrealistic to extrapolate heat penetration data to different

process conditions. Berry and Bush [8] suggested safe

procedures (which might give though rise to quality

problems) for extrapolating data to process with different

retort or initial product temperatures.

The fc = fh Case

A direct application of the broken-line heating curves

analysis is for the case where the slopes of the heating and

cooling curve are different. For such cases, the correlation

between the F value and the fh/U parameter is given by:

fhU¼ qfh þ ð1� qÞfc

FzTref

Fið26Þ

The existence in Ball’s formula method of the q vs g

correlations allows easy handling of the fc = fh case,

through the above equation.

Non Constant Retort Temperature

The equations presented so far assumed constant retort

temperature during heating, TRT, and during cooling, TCW.

Except of some cases of continuous processing, there is

some time period needed for the retort to achieve the con-

stant operating temperature during the heating phase, the so

called coming-up time (CUT). Based on experimental

observations, Ball concluded that a process at constant

retort temperature of duration equal to 42% of the CUT was

equivalent to the process at the variable retort temperature

profile during the CUT. So he suggested to correct the

heating time, by shifting the zero heating time axis by

0.58�CUT and apply his method, as for a process with

CUT = 0, based on this corrected ‘‘zero’’ axis. This cor-

rection factor has been widely used [31]. Obviously, the

percentage of the CUT that is lethal depends upon a number

of parameters, including the rate of change of retort tem-

perature until it reaches its operating value, as well as a

number of product characteristics and kinetic parameters of

the target microorganism [2, 7, 22, 39, 53, 56–58].

No similar correction was proposed for the time needed

for the retort to reach the cooling medium temperature.

Nevertheless, the assumption that retort reaches cooling

medium temperature, TCW, instantaneously after steam-off,

is a conservative approach.

Furthermore, no provisions were made in Ball’s formula

method for variable retort temperature processing. This

makes the method inadequate to compensate for possible

deviations, in terms of retort temperature, during

processing.

The jc Value of 1.41

Using a constant value for the cooling, lag factor jc is

probably the strictest assumption that Ball made in the

8 Food Eng Rev (2010) 2:1–16

123

development of his formula method. As Ball [3] pointed

out: ‘‘… j, generally has a value different from 1.41. It is as

a rule less than this value.’’ In a next paragraph, he

somewhat implied to treat straight-line cooling curves, for

cases where jc = 1.41, as broken-line cooling curves. A

number of investigators, including Ball and Olson’s [6]

presentation of a new method, proposed equations and

presented tables allowing for variable jc values [18–20, 23,

25, 30, 33, 36–38, 51, 52].

With all other parameters being the same, using a jcvalue equal to 1.41 will be a conservative approach if the

true jc value is greater than 1.41. The opposite holds when

the true jc value is less than 1.41. A jc value of unity means

no lag in product temperature at the critical point, indi-

cating a uniform product temperature, as for the case of

complete mixed forced convection heated products. In

general, jc values less than 1 are unrealistic for the critical

point of the product.

A comparison of the effect of cooling jc values on the

fh/U vs log(g) relationship is presented in Fig. 4 with data

obtained from Stumbo [51] for z = 18�F. On the same

figure, Ball’s data for two m ? g values (130 and 180�F)

are included. Note that Stumbo [51] used a single,

constant m ? g value of 180�F to create his tables. Ball’s

values, for either m ? g value, are close to Stumbo’s data

for jc = 1.40, with Ball’s values producing slightly con-

servative results. Differences in Stumbo’s data for jcvalues of 1.00 and 1.40 are rather small, such as Stumbo’s

data for jc = 1.00 are reasonably approached by Ball’s

values. This can give an explanation of the successful

application of Ball’s method to forced convection heating

products. On the same line of thinking, the relative small

deviations between Ball’s and General method process

time calculations for thin profile packages reported by

Ghazala et al. [17] can be attributed to the jc values

involved in these cases, which were about 1.3, a value

close to 1.41, the one used by Ball. Differences in the

fh/U vs log(g) data observed between jc values of 2.00 and

1.40 produce conservative, in terms of safety, results

when Ball’s method is used. This though leads to over-

processing which can be detrimental to product quality.

Similar results were also obtained for all the other

z values used by Ball.

Temperature Rise After Steam-Off

As indicated earlier, the equation used to describe the

initial part of the cooling curve (Eq. 9) does not allow for

any product temperature rise at the beginning of cooling,

which is not valid for the critical point of conduction

heated canned foods (e.g., [13]. In such cases considerable

overprocessing can occur. As indicated in an earlier

publication [47], results obtained by Ball’s formula method

were closer to reference General Method values (in terms

of Fprocess calculations) for conduction heating products,

when the maximum product temperature, rather than the

product temperature at the end of heating, was considered

as Tg. As indicated then, Ball [3] actually defined g as ‘‘the

difference in degrees between the retort temperature and

the maximum temperature attained by the center of a can

during its processing’’. Similar definitions Ball gave in

subsequent publications [4, 6]. From the other hand, by the

definition of heating (process) time through Eq. 21, Ball

considered end of heating when the maximum product

temperature has been reached. This can be confusing, with

the recommendation to Ball’s formula method users being

to consider Tg as the critical point temperature at the end of

heating, allowing for any temperature rise after that to be

added as a safety factor [47]. The use of lower Frequired

values for larger cans of conduction heating foods is

believed to be an attempt to overcome the inability of

Ball’s method to allow for any temperature rise at the

critical point after steam-off [47].

0.1

1

10

100

1000

-1 -0.5 0 0.5 1 1.5 2

log(g) (g in °F)

fh/U

Ball m+g = 180°F

Ball m+g = 130°F

Stumbo jc = 0.40

Stumbo jc = 1.00

Stumbo jc = 1.40

Stumbo jc = 2.00

Fig. 4 Effect of cooling jc values on the fh/U vs log(g) relationship

for z = 18�F (based on Ball and Olson’s and Stumbo’s [6, 51] data)

Food Eng Rev (2010) 2:1–16 9

123

Product Temperature at the Early Stage of Heating

Contrary to the approach that Ball followed to describe the

cooling curve, that is, the use of a separate equation to

describe the early phase of cooling (where product tem-

peratures are high), Ball used a single expression (Eq. 4)

for the entire heating curve. Equation 4 largely underesti-

mates product temperature at the early stage of heating.

However, for most conventional thermal processes, initial

product temperature is low enough so that there is no

destruction taking place at the beginning of the heating

and, therefore, the error in using Eq. 4 instead of a more

accurate one [20] is negligible. This might be though of

concern for aseptic processes [11] where high initial

product temperatures are encountered.

Product Temperatures 80�F Below TRT

As limits of integration in Eq. 3, Ball used a critical point

temperature 80�F below the retort temperature in both the

beginning of heating and at the end of cooling. He assumed

that there is no lethal effect of the process for product

temperatures less than 80�F below the retort temperature.

This is in general a valid assumption and can be prob-

lematic only if high z values are used. For z values up to

26�F that Ball used, this does not pose any problems.

Parametric Values

Ending our discussion concerning the validity of the

assumptions Ball made in developing his method, we must

emphasize the need of accurate knowledge of the various

parameters involved in the above calculations (for example

fh, jh, and z values). Deviation from first order kinetics and

the z value concept, as far as the effect of temperature on

the decimal reduction time is concerned, will require

reevaluation of Eq. 10 and therefore the whole method.

First order kinetics can be seen as a subset of different

inactivation schemes [59] and presumably, any thermal

process calculation method that is based upon such a model

will have increased flexibility and applicability. Statistical

variation in physical, reaction kinetics and operational

parameters can lead to large deviations in Fprocess, Frequired,

and heating time calculations. A comprehensive discussion

on this matter is given by [24].

Use of Algebraic Equations to Replace Ball’s Graphs

The use of Ball’s method relies on the tables or the graphs

with the discrete values of the solution of Eq. 10 that Ball

presented in the form of fh/U vs log(g) relationships. The

use of graphs or tables is in general difficult and susceptible

to errors. Several investigators tried to simplify the use of

the method through nomograms [35], computerized pro-

cedures [40] algebraic equations [60] or artificial neural

networks [1]. Use of algebraic equations in place of tabu-

lated fh/U vs log(g) values can ease the use of any thermal

process calculation method [26]. The polynomial equations

used by Vinters et al. [60] were though restricted to a z

value of 18�F. Thus, a new set of working, regression

equations were developed in order to facilitate the appli-

cation and use of Ball’s method, and they will be presented

in the rest of this paper.

The use of dimensionless variables can greatly reduce the

amount of tables, graphs or equations needed to express a

given relationship. Hayakawa [20] used a ratio of z values as

a working variable, Steele and Board [44] used ‘‘sterilization

ratios’’, that is, ratios of temperature differences between

product and heating (or cooling) medium over z values, while

[36, 37] used g/z ratios in his correlations for thermal process

calculations. A careful look at Eq. 10 reveals that tempera-

ture ratios of g/z and m/z are involved. If log(g/z) instead of

log(g) is used in the abscissa of the fh/U vs log(g) plots

(Fig. 3), one can see that curves for different z values almost

coincide. If a further transformation of the form of log(g/z)-

z/zc is used, curves for different z values become even closer,

as it is shown in Fig. 5 for the m ? g = 180�F data (same

data used for Fig. 3). For Fig. 5, a value of zc equal to 430�F

was used for demonstration purposes. A more accurate value

can be determined through a regression analysis and an

appropriate model.

Based on the above observation, an algebraic equation

correlating log(fh/U) (note the logarithmic scale used in

Fig. 3 and 5) with log(g/z)-z/zc was sought. After several

trials, the following equation is proposed to replace Ball’s

tabulated fh/U vs g values:

y ¼ a1

1þ a2e�a3xþ a4

1þ a5e�a6xþ a7 ð27Þ

The choice of Eq. 27 was made based on the residual

sum of squares error (SSE) and the correlation coefficient

(R2) between predicted and tabulated values (Table 1), and

the behavior of the proposed equation at the limits of the

variables and the parameters involved. Variables x and y

correspond to either log(fh/U) or log(g/z)-z/zc and vice

versa (Table 1) depending if we are seeking an fh/U value

for a given g, or the opposite. The nonlinear nature of the

correlation (Eq. 27) requires explicit algebraic equations

for both fh/U and g. The coefficients of Eq. 27 were

determined through a nonlinear regression using the

tabulated fh/U vs g data given by Ball and Olson [6] for

two m ? g values of 130�F and 180�F (Table 1). Although

data for m ? g = 160�F were also given in the original

publication [3], the fact that the m ? g parameter only

slightly affects the fh/U vs g relationship left us the option

10 Food Eng Rev (2010) 2:1–16

123

of using only the two m ? g edge values. For every 10�F

change in the m ? g values, Stumbo [51] reported only an

error of 1% in the F value. An indication of the error

magnitude involved due to the m ? g variation can be

inferred from Fig. 4 by comparing fh/U vs g values for

m ? g = 130�F and m ? g = 180�F for z = 18�F. Note

that the use of the m ? g = 180�F data for any m ? g

value of less than 180�F represents a conservative

approach. Interpolation between the values obtained

through Eq. 27 for m ? g = 130�F and m ? g = 180�F

is recommended for intermediate m ? g values.

Note that the last term a7 in Eq. 27 was not necessary to

express the log(g/z) vs log(fh/U) relationship. All coeffi-

cients a1 through a7 appearing in Eq. 27 are dimensionless

with the exception of the zc coefficient which has units of

temperature difference, as the regular z value does. Thus,

temperature units in degrees Celsius can be used in Eq. 27

as long as the zc values presented on Table 1 will be

converted to temperature difference in degrees Celsius by

dividing the values given in Table 1 by 1.8.

Comparisons between predicted, through Eq. 27 and the

coefficients presented in Table 1, and Ball’s tabulated fh/U

vs log(g) data for m ? g = 180�F are presented in Fig. 6.

The agreement between the two data sets was very good.

The percent relative error between predicted and Ball’s

tabulated fh/U values was, with few exceptions,

within ±4%, and in the majority of the cases within ± 2%

(Fig. 7). This relative error in the fh/U values directly

reflects the relative error between Fprocess values calculated

through Eq. 27 or Ball’s tabulated fh/U vs log(g) data.

Similar results were obtained for the rest of the z values

for which Ball’s tabulated data were available, as well as

for the m ? g = 130�F data.

For the reverse calculations, that is, log(g) vs fh/U,

comparisons between predicted and Ball’s values revealed

also very good agreement, as shown for the m ? g =

130�F data which are indicatively presented in Fig. 8.

Absolute errors between predicted and Ball’s tabulated

log(g) values were less than ± 0.02 with the majority

being less than ± 0.01 (Fig. 9). In view of Eq. 21, the

absolute error in log(g) values is transferred through the

0.1

1

10

100

1000

-2.5 -2 -1.5 -1 -0.5 0 0.5

log(g/z)-z/zc

f h/U

z = 6°F

z = 10°F

z = 14°F

z = 18°F

z = 22°F

z = 26°F

Fig. 5 Re-plot of Fig. 3 data using a dimensionless log(g/z)-z/zc

variable (a zc value equal to 430�F is used)

Table 1 Values of the

coefficients of Eq. 27 according

to the definitions of the x and yvariables

m ? g = 130�F m ? g = 180�F

y log(g/z)-z/zc log(fh/U) log(g/z)-z/zc log(fh/U)

x log(fh/U) log(g/z)-z/zc log(fh/U) log(g/z)-z/zc

a1 -0.088335831 40.122199 -3.3545727 22.016510

a2 -0.96375429 38.533071 -0.34453049 21.598294

a3 0.028257272 2.3715954 0.42100067 2.4586869

a4 1.0711536 5.3058320 4.0057210 38.202986

a5 0.19518983 2.8885491 0.13211471 23.706331

a6 4.5699218 0.63534158 3.2971998 0.49435142

a7 - -0.63814873 – -0.74859566

zc (�F) 389.10600 405.49832 389.48491 468.11021

R2 0.999932 0.999951 0.999924 0.999954

SSE 0.0202 0.0202 0.0215 0.0185

Data points 584 584 578 578

Food Eng Rev (2010) 2:1–16 11

123

fh value as an absolute error to the required heating time

calculations. Thus, a 0.01 absolute error in log(g) generates

a -0.01 9 fh absolute error in the heating time, B.

The use of Eq. 27 can largely facilitate Ball’s formula

method calculations. It cannot only make calculations easier

and with minimal error (compared to graphical data) but it

allows calculations for intermediate values of the parameters

involved; that is, for any intermediate fh/U and g (compared

to the tabulated data) as well as z values. It is worth men-

tioning that Eq. 27 produced reasonable predictions beyond

the z value of 26�F, which was the limit of Ball’s data, due to

its dimensionless form and the reasoning behind its selection

from a number of other regression equations. So, for

example, fh/U vs g values for z = 54�F through Eq. 27 were

almost identical to Stumbo’s [51] values for jc = 1.40.

Without recommending extrapolation of data, the above

observation is to support the value of Eq. 27.

Equation 27 is based on the tabulated fh/U vs g data

given by [6] where g values greater of about 0.1�F were

used. So, Eq. 27 is only valid for g C 0.1�F. For cases

where g is less than 0.1�F, Eq. 19 or 20 must be used

(either for fh/U or log(g)) with the fhU

��g¼0:1

or fhU

��g¼g0

values

needed for such calculations to be taken from Ball’s tables,

or determined through Eq. 27.

Ending our presentation in the use of algebraic equations

to replace Ball’s graphs and tables, and for the complete-

ness of calculations, the following equation, Eq. 28, was

developed to replace q vs g tabulated or graphical data

needed for broken-heating curves and/or the cases where

fc = fh.

q ¼ b1

1þ b2e�b3 logðgÞ þ1

1þ b4e�b5 logðgÞ þ ðb6gþ b7Þz

ð28Þ

Based on Ball and Olson [6] q vs g tables, the

coefficients of Eq. 28 were estimated and are presented

0.1

1

10

100

1000

-1 -0.5 0 0.5 1 1.5 2

log(g) (g in °F)

fh/U

z = 6°F

z = 10°F

z = 14°F

z = 18°F

z = 22°F

z = 26°F

Fig. 6 Comparison between predicted, through Eq. 27 and the

coefficients presented in Table 1, (lines) and Ball’s tabulated fh/Uvs log(g) data (open cycles) for m ? g = 180�F

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

00010010111.0

f h /U Ball

(fh/

UB

all-

f h/U

pred

icte

d)/

(fh/

Uba

ll)

(%

)

z = 6°F z = 10°F z = 14°F

z = 18°F z = 22°F z = 26°F

Fig. 7 Relative error between

predicted, through Eq. 27 and the

coefficients presented in Table 1, and

Ball’s tabulated fh/U data for

m ? g = 180�F

12 Food Eng Rev (2010) 2:1–16

123

in Table 2, for m ? g = 130, 160 and 180�F. A

comparison between predicted, through Eq. 28 and the

coefficients presented in Table 2, and Ball’s tabulated q vs

g data for m ? g equal to 130 and 180�F are indicatively

presented in Fig. 10. As it can be seen (Fig. 10), the

agreement between the predicted and Ball’s data was very

-1

-0.5

0

0.5

1

1.5

2

0.10 1.00 10.00 100.00 1000.00

f h /U

log(

g) (

g in

°F

)

z = 26°F

z = 22°F

z = 18°F

z = 14°F

z = 10°F

z = 6°F

Fig. 8 Comparison between predicted, through Eq. 27 and the coefficients presented in Table 1, (lines) and Ball’s tabulated log(g) vs fh/U data

(open cycles) for m ? g = 130�F

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

-1.0 -0.5 0.0 0.5 1.0 1.5

log(g) Ball (g in °F)

log(

g)B

all-

log(

g)pr

edic

ted

z = 6°F z = 10°F z = 14°F

z = 18°F z = 22°F z = 26°F

Fig. 9 Absolute error between predicted, through Eq. 27 and the coefficients presented in Table 1, and Ball’s tabulated log(g) data for

m ? g = 130�F

Food Eng Rev (2010) 2:1–16 13

123

good. The maximum relative error between predicted and

Ball’s tabulated q values reported in Table 2, is rather

unjust for the correlation presented by Eq. 28, usually due

to some accidental extreme values.

Conclusions

Ball’s original formula method for thermal process calcu-

lations continues to serve the food industry since its

development in 1923. It represents an excellent example of

the intelligence use of mathematics in food processing.

In the preceding paragraphs, the fundamental ideas

behind thermal process calculations were initially out-

lined. Thereafter, the steps involved in the development

of Ball’s method were given with the appropriate clari-

fications. Key assumptions associated with the method

and their implications on product safety and quality were

discussed, and the conservative nature of the method was

pointed out.

Finally, a set of algebraic equations to replace Ball’s

fh/U vs log(g) tabulated or graphical data for g values greater

than 0.1�F, Eq. 27, and q vs log(g) data, Eq. 28, were pre-

sented. The use of these equations introduced negligible

error, can greatly facilitate the use of Ball’s original formula

method, and permit calculations for intermediate values of

the parameters involved. An explicit expression (Eq. 19 or

20) according to Ball’s assumptions, for the fh/U vs g rela-

tionship for g less than 0.1�F, was also given.

References

1. Afaghi MK (2000) Application of artificial neural network

modeling in thermal process calculations of canned foods. MSc

Thesis, Dept Food Science and Agr Chem, McGill University,

Montreal, Canada

2. Alstrand DV, Benjamin HA (1949) Thermal processing of canned

foods in tin containers. V. Effect of retorting procedures on

sterilization values in canned foods. Food Res 14:253–257

3. Ball CO (1923) Thermal process time for canned food. Bulletin

of the National Research Council No. 37., 7, Part 1Natl Res

Council, Washington, DC

4. Ball CO (1928) Mathematical solution of problems on thermal

processing of canned food. Univ Calif Pubs Public Health, vol 1,

No 2. University of California Press, Berkeley, CA (with sup-

plement in 1936)

5. Bal CO (1949) Process evaluation. Food Technol 3:116–118

6. Ball CO, Olson FCW (1957) Sterilization in food technology.

Theory practice and calculations. McGraw-Hill Book Co., New

York

7. Berry MR Jr (1983) Prediction of come-up time correction factors

for batch-type agitating and still retorts and the influence on

thermal process calculations. J Food Sci 48:1293–1299

8. Berry MR Jr, Bush RC (1987) Establishing thermal processes for

products with broken heating curves from data taken at other

retort and initial temperatures. J Food Sci 52(4):958–961

9. Bigelow WD, Bohart GS, Richardson AC, Ball CO (1920) Heat

penetration in processing canned foods. Bull. No. 16-L, Res. Lab.

Natl Canners Assoc, Washington, DC

10. Cleland AC, Robertson GL (1985) Determination of thermal

processes to ensure commercial sterility of foods in cans. In:

Thorne S (ed) Developments in food preservation, vol 3. Elsevier

Applied Science Publishers, London

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 1 10 100

log(g) (g in °F)

ρ

6°F 14°F 18°F 26°F

6°F 14°F 18°F 26°F

m + g = 180°F

m + g = 130°F

Fig. 10 Comparison between predicted, through Eq. 28 and the

coefficients presented in Table 2, (lines) and Ball’s tabulated q vs

log(g) data (open symbols) for m ? g = 130�F and for

m ? g = 180�F

Table 2 Values of the coefficients of Eq. 28

m ? g = 130�F m ? g = 160�F m ? g = 180�F

b1 0.289514 0.0552200 0.00993234

b2 -4.87991 -1.84861 -1.16901

b3 0.553369 0.240032 0.0647075

b4 0.0287250 0.0314127 0.0329320

b5 -1.89101 -1.79959 -1.72143

b6 -0.0000302250 -0.0000244004 -0.0000198010

b7 -0.000814476 -0.000521808 -0.000448516

R2 0.999928 0.999940 0.999958

SSE 0.000599 0.000362 0.000211

Data points 220 220 220

Maximum

relative error

2.65% 1.43% 0.52%

14 Food Eng Rev (2010) 2:1–16

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